Welcome to Geometry & Trigonometry — the module where numbers meet shapes. Whether it is finding the distance between two points, writing the equation of a circle, or solving a triangle with the Law of Cosines, this module gives you the tools to work with space and angles.
You will start with coordinate geometry (points, lines, distances), progress through circles, ellipses, and hyperbolas, explore the unit circle and trigonometric identities, and finish with vectors. These topics form a significant portion of the CSCA Math exam and connect naturally to physics and engineering.
Grab your protractor (just kidding — you will not need one!) and let's begin.
Geometry and trigonometry typically account for 6-8 questions worth 25-35 marks on the CSCA exam. Expect coordinate geometry calculations, circle equation problems, triangle-solving with Laws of Sines/Cosines, conic section identification, and vector computations. Draw diagrams, use exact values, and show your working clearly.
Coordinate geometry (analytic geometry) places geometric shapes on the xy-plane so we can study them with algebra.
Key Formulas
d = √((x₂-x₁)² + (y₂-y₁)²). This is just the Pythagorean theorem applied to the horizontal and vertical differences.M = ((x₁+x₂)/2, (y₁+y₂)/2). Average the coordinates.m = (y₂-y₁)/(x₂-x₁). Rise over run.Gradient Relationships
Collinear Points
Three points are collinear (on the same line) if the gradient between any two pairs is equal.
Section Formula
The point dividing the segment from (x₁,y₁) to (x₂,y₂) in the ratio m:n is: ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).
These basics underpin every geometry question. The distance formula alone can determine whether a triangle is isosceles, right-angled, or equilateral.
💡Distance = √((Δx)²+(Δy)²). Midpoint: average coordinates. Gradient: rise/run. Parallel: equal gradients. Perpendicular: m₁·m₂ = -1.
📋 Key Formulas
d = √((x₂-x₁)²+(y₂-y₁)²)M = ((x₁+x₂)/2, (y₁+y₂)/2)m = (y₂-y₁)/(x₂-x₁)
📝 Worked Example 1
Example 1: Find the distance between A(1, 3) and B(4, 7).
d = √((4-1)²+(7-3)²) = √(9+16) = √25 = 5.
📝 Worked Example 2
Example 2: Find the midpoint of (-2, 6) and (4, -2).
M = ((-2+4)/2, (6+(-2))/2) = (1, 2).
📝 Worked Example 3
Example 3: Line L₁ has gradient 2. Find the gradient of a line perpendicular to L₁.
m₂ = -1/m₁ = -1/2.
🧠Label your points clearly as (x₁,y₁) and (x₂,y₂) before substituting into formulas.
🧠For "show that the triangle is right-angled", compute all three side lengths and check if a² + b² = c².
⚠️Subtracting coordinates in wrong order: Be consistent — (x₂-x₁) and (y₂-y₁) must use the same order.
⚠️Forgetting the square root: d² = (Δx)²+(Δy)² — do not forget to take the root for the actual distance.
🎯 Try This Yourself
Find the distance between (0, 0) and (5, 12).
In this module you explored coordinate geometry (distances, lines, circles), mastered trigonometry (unit circle, identities, Laws of Sines and Cosines), studied conic sections (ellipses and hyperbolas), and learned vector operations. These tools let you analyse shapes, solve triangles, and describe curves — skills that appear throughout the CSCA exam and in physics, engineering, and computer science.
Open and read all sections to complete this module